Where Does the Noise Come From? A Variance-Components Decomposition of Non-Determinism in LLM Brand Answers

arXiv:2607.13304v1 Announce Type: cross Abstract: Teams measuring whether large language models (LLMs) recommend a brand face a reproducibility problem: ask the same question twice and the answer moves. Practice resamples each prompt a few times (commonly five) and averages, treating within-prompt resampling as the source of the noise. But a measured brand score moves for at least four separable reasons: within-prompt resampling, prompt paraphrase, model identity, and query language. We specify a crossed random-effects (generalizability-theory) decomposition that partitions the total variance of a response-level brand outcome into these four sources, and embed the components in a decision-study allocation that returns how many repeats, paraphrases, models, and languages to buy for a target reliability. We apply it to a fully crossed corpus of 12,933 LLM responses on 20 Central and Eastern European brands, 8 languages, and 3 models (GPT-5.2 and Gemini 3 Flash in parametric mode, Perplexity in grounded retrieval), with a stability subset of 1,435 cells resampled about five times. The outcome is per-response multilingual sentiment polarity. Query language is the largest systematic facet (26.5% of the variance of one response) against 1.5% for brand identity (ICC 0.0146), so a single AI answer carries almost no brand-discriminating signal. Once a cell term isolates pure resampling, resampling is 34.8% of variance and the brand-in-context interaction 29.6%; brand-by-language is 8.6% (a bilingual penalty) while brand-by-model and brand-by-prompt are near zero. Per unit of query budget, adding languages and models reduces relative-error variance far more than adding repeats: a repeat past the fifth reduces it by only 0.0003. Brand-ranking reliability stays low, near 0.01 for a single answer and about 0.36 at the full crossed design, so reliability is bought by spreading across languages and models, not by repeating one prompt.